Session ids are stored in cookies. Cookies are handled by libsoup in this case. I think by default cookies are kept just in memory and not stored in a file. It should be possible to change it. I will have a look.
A question remains: should we store cookies in a file while there is no way of deleting them from the browser?
Are cookies preserved through sessions on default Kindle browser?

maybe shove them in /var (after sanity testing for size) and at least in that way they would live until a power cycle?

maybe set aside a fixed maximum (say 5Mb?) on the FAT - that should hold a baziilion cookies... In a folder called "COOKIES-STORED-HERE" or something That way they could be easily hosed manually.

anyways - starved of caffeine and with almost no actual cerebral function to speak of today - those are my thoughts.